"Menelaus: an unnecessary requirement in a proof?"

Hello Alex,

I have a comment concerning a proof of Menelaus's Theorem that is presented during the discussion of Einstein's remarks on elegant and ugly proofs (https://www.cut-the-knot.org/Generalization/MenelausByEinstein.shtml#Carnot). It is the proof that is marked "Proof #3" in the corresponding applet, and which serves as an impressive example of a proof which is elegant despite involving auxiliary constructions...

The proof starts "Draw a line perpendicular to the transversal EDF ...". My comment is this: the proof does not seem to make, actually, any use of the fact that the auxiliary line (call it l_{1}) is perpendicular to the transversal. The proof does use the following principle: "the segments cut on two lines by a family of parallel lines are in the same ratio". Therefore it is required that the segments aKa, bKb, cKc are all parallel to the transversal. But for that they don't seem to have to be perpendicular to l_{1}. The only requirement from l_{1} seems to be that it intersects the transversal EDF.

1. "RE: Menelaus: an unnecessary requirement in a proof?"
In response to message #0

>Or perhaps I've missed something.

No, that's a good remark. The only thing that matters is that the lines through the vertices come out parallel to the transversal. Surely this might be the starting point with a line crossing the four as the second stage.

2. "RE: Menelaus: an unnecessary requirement in a proof?"
In response to message #1

By the way, when I posted, I marked the check-box to get email notifications "when a new message is submitted". Also for my previous post. But I didn't get any such notifications. Do you know perhaps why? I verified that they weren't held by gmail's spam filter.

4. "RE: Menelaus: an unnecessary requirement in a proof?"
In response to message #2

Thank you for taking the trouble. There is certainly a problem with the forum software. The program is quite old and is no longer supported. I am planning to switch to a new program in a short while and thus resolve several outstanding problems, this one in particular.